A379654 Least positive integer k <= n such that |tau(k)|*n + 1 is prime, or 0 if such k does not exist, where the tau function is given by A000594.

1, 1, 2, 1, 5, 1, 5, 2, 3, 1, 4, 1, 2, 2, 11, 1, 2, 1, 2, 5, 13, 1, 4, 2, 2, 3, 5, 1, 3, 1, 5, 2, 3, 6, 3, 1, 21, 15, 2, 1, 3, 1, 2, 11, 5, 1, 2, 2, 6, 2, 3, 1, 4, 2, 2, 11, 7, 1, 3, 1, 3, 2, 3, 5, 3, 1, 2, 3, 2, 1, 4, 1, 2, 2, 2, 6, 10, 1, 15, 3, 4, 1, 2, 2, 5, 3, 2, 1, 2, 2, 6, 12, 4, 3, 2, 1, 7, 3, 2, 1

Offset

1,3

Comments

Conjecture 1: a(n) > 0 for all n > 0. In other words, for each positive integer n, there is a number k among 1,...,n such that |tau(k)|*n + 1 is prime.

Conjecture 2: For each integer n > 1 not equal to 22, there is a number k among 1,...,n such that |tau(k)|*n - 1 is prime.

We have verified both conjectures for n up to 10^8.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Example

a(1) = 1 since 1*|tau(1)| + 1 = 2 is a prime.
a(5) = 5 since 5*|tau(5)| + 1 = 5*4830 + 1 = 24151 is prime, and 5*|tau(k)| + 1 is composite for every k = 1, 2, 3, 4.

Mathematica

t[n_]:=t[n]=Abs[RamanujanTau[n]];
L={}; Do[Do[If[PrimeQ[t[k]n+1], L=Append[L, k]; Goto[aa]], {k, 1, n}]; L=Append[L, 0]; Label[aa], {n, 1, 100}]; Print[L]

Crossrefs

Cf. A000040, A000594, A034693.

Sequence in context: A318320 A369741 A014652 * A060448 A378543 A090080

Adjacent sequences: A379648 A379649 A379652 * A379655 A379656 A379657

Keyword

nonn,new

Author

Zhi-Wei Sun, Dec 28 2024

Status

approved